Gravity Seven Gaps Freudenthal Torus Class Mass Torus Class Mass Eq Fiber Card M

The class mass identity

In mathematics, when many objects are considered equivalent under a symmetry, a common move is to treat the whole class of them as one thing. The Recognition Science framework's library, a machine-checked collection of formal theorems, does this for a family of geometric objects called bounded complexes. The declaration torus_classMass_eq_fiberCard_mul_mu proves an identity about the class of a particular complex, the Freudenthal torus of size N. It states that the class mass, the total weight assigned to the entire equivalence class, equals the number of labeled objects in that class, called the fiber cardinality, multiplied by the symmetry factor of the canonical torus itself.

The identity is a bookkeeping rule. It says the weight of a whole collection of equivalent objects is not simply the weight of one representative. Instead, you must count how many distinct labeled objects collapse into the same class, and multiply that count by the weight of the single representative. The framework proves this as a theorem, not a definition, and the proof is axiom-clean, meaning it uses no special assumptions beyond the standard logical foundations. The theorem is the honest form of a claim that was previously worded incorrectly, and the module records that repair explicitly.

What the theorem does not claim is just as important as what it proves. It does not say that the class mass is bounded by 1 divided by N cubed. That statement would be false in general, because the fiber cardinality grows as N grows, and the theorem does not bound that growth. The framework's own status ledger marks the absolute suppression claim as false, a killed claim. The theorem also does not claim that the class mass sequence converges to zero, nor does it make any statement about a continuum limit or about the existence of a derived measure on a substrate. Those remain open targets, flagged red in the status record.

The distinction matters for the framework's broader program. A labeled object, one specific representative, has its mass bounded by 1 over N cubed, and that bound is proved. But the pushforward class mass, the weight of the whole class, carries the extra fiber factor, and that factor is not controlled. Anyone who wants to use the class mass in a limit argument must confront this growth. The theorem forces that confrontation into the open, which is exactly what a consistency gate is for.

/-- **THEOREM (class mass, identity form).**  The pushforward class mass
of the torus class is the labeled fiber cardinality times the symmetry
factor of the canonical torus:
`classMass(⟦T_N⟧) = |fiber(⟦T_N⟧)| · μ(T_N)`. -/
theorem torus_classMass_eq_fiberCard_mul_mu (N : ℕ) [NeZero N] :
    classMass (Quotient.mk (relabelSetoid (7 * N ^ 3))
        (freudenthalBoundedComplex N)) =
      (fiberCard (relabelSetoid (7 * N ^ 3))
          (Quotient.mk (relabelSetoid (7 * N ^ 3))
            (freudenthalBoundedComplex N)) : ℝ) *
        mu (freudenthalBoundedComplex N) := by
  rw [classMass_eq_fiberCard_mul_mu]
  congr 1
  exact mu_congr (equivalent_of_mk_eq (Quotient.out_eq _))

/-- **THEOREM (class mass, identity form).**  The pushforward class mass
of the torus class is the labeled fiber cardinality times the symmetry
factor of the canonical torus:
`classMass(⟦T_N⟧) = |fiber(⟦T_N⟧)| · μ(T_N)`. -/
theorem torus_classMass_eq_fiberCard_mul_mu (N : ℕ) [NeZero N] :
    classMass (Quotient.mk (relabelSetoid (7 * N ^ 3))
        (freudenthalBoundedComplex N)) =
      (fiberCard (relabelSetoid (7 * N ^ 3))
          (Quotient.mk (relabelSetoid (7 * N ^ 3))
            (freudenthalBoundedComplex N)) : ℝ) *
        mu (freudenthalBoundedComplex N) := by
  rw [classMass_eq_fiberCard_mul_mu]
  congr 1
  exact mu_congr (equivalent_of_mk_eq (Quotient.out_eq _))

/-- **THEOREM (the honest class-mass bound).**
`classMass(⟦T_N⟧) ≤ |fiber(⟦T_N⟧)| / N³`.  The fiber cardinality is NOT
bounded here; in particular this does NOT yield the (killed) absolute
`N⁻³` suppression of the pushforward mass. -/
theorem torus_classMass_le_fiberCard_div_cube (N : ℕ) [NeZero N] :
    classMass (Quotient.mk (relabelSetoid (7 * N ^ 3))
        (freudenthalBoundedComplex N)) ≤
      (fiberCard (relabelSetoid (7 * N ^ 3))
          (Quotient.mk (relabelSetoid (7 * N ^ 3))
            (freudenthalBoundedComplex N)) : ℝ) / ((N : ℝ) ^ 3) := by
  rw [torus_classMass_eq_fiberCard_mul_mu, div_eq_mul_one_div]
  exact mul_le_mul_of_nonneg_left (mu_freudenthal_le_inv_cube N)
    (Nat.cast_nonneg _)

What this page does not claim

The pushforward class mass is absolutely suppressed as 1 over N cubed. The class mass sequence converges to zero as N grows. Any continuum limit or derived substrate measure exists for the torus class.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/SevenGaps/FreudenthalTorusClassMass.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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